\newproblem{lay:4_3_33}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 4.3.33}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Consider the polynomials $p_1(t)=1+t^2$ and $p_2(t)=1-t^2$. Is $\{p_1(t),p_2(t)\}$ a linearly independent set in $\mathbb{P}_3$? Why or why not?
}{
  % Solution
	We may define the linear transformation $T:\mathbb{P}_3\rightarrow\mathbb{R}^4$ such that $T(a+bt+ct^2+dt^3)=(a,b,c,d)$. It can be easily verified that
	$T$ is a linear transformation.
	
	The polynomials $p_1(t)$ and $p_2(t)$ are transformed to
	\begin{center}
		$T(p_1(t))=(1,0, 1,0)$ \\
		$T(p_2(t))=(1,0,-1,0)$ \\
	\end{center}
	which is clearly a linear independent set in $\mathbb{R}^4$ and by Exercise Lay 4.3.31, this implies that $\{p_1(t),p_2(t)\}$ is a linearly independent
	set in $\mathbb{P}_3$.
}
\useproblem{lay:4_3_33}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
